Problematics | As the crow flies
Centuries ago, one dropped pebbles into a jar to be able to drink water. This week, another smart crow is looking for an optimum flight path
Once upon a time, a thirsty crow came across a jar in which the water was too low for its beak to reach. You know what happens next, pebbles and all. In fact, you may also have seen such a situation play out outside of the fable: there is at least one video on social media showing a crow doing exactly what its fabulous counterpart did.
Crows are indeed smart birds, creatures you would expect to use their mathematical and scientific acumen in other situations that demand it. Come, let’s meet another smart one.
#Puzzle 82.1
A large garbage dump lines one side of a long straight road from one end to the other. Any piece of garbage is food for a scavenger like the crow, so you can assume that any point it lands on has a piece of food that our crow would like to carry to her nestlings.
The crow’s nest is on a tree on the other side of the road. At the moment our puzzle begins, the crow is on another tree that is on the same side of the road as the tree on which her nest rests, but there is some distance between the two trees.
Our crow wants to go back home with food for the kids. For that, she needs to cross the road by flying to the ground, pick up any item of garbage at any point, and cross the road back to her side of the street and reach her nest. Although any point on the other side has the food she needs, she wants to keep this flight as short as possible, i.e., she wants to travel the shortest possible distance.
From which point on the opposite side should the crow pick up the food so that its flight path is the shortest possible?
#Puzzle 82.2
Take all the digits 1 through 9 and observe this little curiosity:
6729/13458 = 1/2
All 9 digits reducing to a neat fraction. There may be other such arrangements leading to 1/2, but that is not the puzzle.
Can you use all 9 digits to form fractions that reduce to 1/3, 1/4, 1/5, 1/6, 1/7, 1/8 and 1/9?
MAILBOX: LAST WEEK’S SOLVERS
#Puzzle 81.1
Hi Kabir,
Let a and b be the sides of the first plot, a > b.
Let c and d be the sides of the second plot, c > d.
Let p = cd and q = a + b, then ab = 3p and c + d = 3q.
Exploring, we get many solutions. Some of these are -
a = 22, b = 15, c = 110, d = 1
a = 35, b = 12, c = 140, d = 1
a = 48, b = 11, c = 176, d = 1
a = 42, b = 31, c = 217, d = 2
— Professor Anshul Kumar, Delhi
I have edited out significant (and interesting) portions of the above letter. Professor Kumar determines ranges within which the possible answers must fall, but those descriptions are too elaborate to reproduce here. Some other readers too have described methods (among them, Yadvendra Somra has mailed me a number of times with an improved method each time). Others have just gone to one or more possible answers by using hit and trial.
#Puzzle 81.2
Hi Kabir,
The floriculturist made a loss of ₹10,000 if we ignore the notional profit loss of ₹20,000 (the profit is notional, there is no real money lost). The breakup of ₹10,000 is ₹9,000 of effort and material plus the ₹1000 given in cash to the customer. The Problematics writer and Problematics solver each loan ₹30,000 and get back ₹30,000, hence "hisab kitab barabar".
— Akshay Bakhai (Mumbai)
Solved both puzzles: Professor Anshul Kumar (Delhi), Akshay Bakhai (Mumbai), Yadvendra Somra (Sonipat), Dr Sunita Gupta (Delhi), Shishir Gupta (Indore), YK Munjal (Delhi), Group Captain RK Shrivastava (retired, Delhi)
Solved #Puzzle 81.2: Dr Vivek Jain (Baroda), Ajay Ashok (Mumbai)
